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Solution Manual Chapter 6: Aeronautical
Engineering
1. Complete the following table where
1i
.
i-3
i-2
i-1
i0
i1
i2
i3
i4
i-5
i6
i
-1
-i
1
i
−1
−i
1
i
−1
2. Expand (1 + i)2 = ? (Answer: 2i)
3. Carry out the multiplication (3 + 2i)(1 + 7i) = ? (Answer: −11 + 23i)
4. What is (a + bi)(a − bi) = ? (Answer: a2 + b2)
5. The absolute value of a complex number is defined to be the distance from the origin
to the number in the Argand plane using the Pythagorean Theorem. What is the
absolute value of z = x + iy? (Answer: the absolute value of z is
22
xy
)
i
4
11
27 i
i
7
i
7
i
7
50
50
50
149
7. Simplify:
i
i
31
23 2
.
10
27i31
i
i
i
i
i
i
i
ii
i
i
31
31
31
125
31
31
31
4129
31
23 2
2
8. What are the real and imaginary parts of A = 1/z2? Write your answer in the form
Write x2 - y2 as F to simplify the algebra:
22
2
2222
2
22
22
22222222222
4
2
4
4
2
44
2
4
2
22
2
yxyx
xyi
yxyx
yx
Z
yxF
xyi
yxF
F
yxF
xyiF
xyiF
xyiF
xyiFxyiF
xyiF
9. Look up Euler’s/de Moivre’s Theorem in your high school mathematics text book (or
the Interenet). It says that ei
= cos + isin where is the rotational angle in the complex
plane1. Draw lines corresponding to = 0, /4, /2, 3/4 and . What are their x, y
values on the surface of a unit circle?
1 You can see that the translation between polar coordinates and Cartesian coordinates is easy using
standard trig functions, e.g., x = R cos = cos and y = R sin = sin if R = 1 etc.
150. m/s if the air density is 0.525 kg/m3 and the circulation is 55.0 m2/s?
Need: Total lift on a 21.0 m span airfoil.
Know: The airfoil is travelling at 150. m/s, in air at a density of 0.525 kg/m3.
Circulation is 55.0 m2/s. Acceleration due to gravity is 9.81 m/s2.
11. In Example 6.8, the SFC for a Boeing 787 is 0.587 lbm/hr/lbf leading to a fuel
consumption rate of 12,680 lbm/hr. If a standard aviation jet fuel (JP- 4) costs $3.50/US
gallon, has a density of 6.84 lbm/US gallon, how much does fuel cost for each hour of
level flight?
Need: Fuel costs for a wide bodied jet.
Know: SFC is 0.587 lbm/hr/lbf leading to a fuel consumption rate of 12,680
lbm/hr. Fuel density is 6.84 lbm/US gallon. One US gallon costs $3.50.
12. An aircraft has a specific fuel consumption SFC of 0.450 lbm/hr per lbf of thrust.
Assume this is a constant during normal steady cruise conditions. The thrust over this
period is a constant 5,120 lbf. On a flight lasting 2.00 hours how much fuel is consumed
by this aircraft? How much is the cost of fuel over this 2 hour flight?
Assume 1.00 lbm of fuel = 0.150 US gallons and that it costs $3.50/US gallon. Ignore the
weight changes due to fuel consumption en-route.
Need: Fuel consumed in a 2.00 hr flight.
Know: SFC = 0.450 lbm/hr/lbf thrust. Fuel costs $3.50 per US gallon.
1.00 lbm of this fuel = 0.150 US gallons; thrust = 5,120 lbf.
13. Most large passenger planes fly at a cruising speed of about 0.85 × Mach 12. The
Mach number is defined as the ratio of actual speed V to the speed of sound, c.
The Mach number is an important variable in most speeding objects moving at or
exceeding the local speed of sound. The speed of sound is temperature sensitive as the √T
with T in kelvin. At 20ºC the speed of sound is 343 m/s or 767 mph.
At an altitude of 10,000 m, the atmosphere has cooled to -45ºC. What is the speed of
sound? (m/s and mph please).
Need: Speed of sound in air at 20ºC and at -45ºC in m/s and mph.
Know: Mach number varies as square root of absolute temperature. At 293K,
Mach 1 = 343 m/s or 767 mph.
2 Ernst Mach was an Austrian physics known especially for his work on shock waves. You have heard a
shock wave when lightning strikes nearby.
14. Refer to Example 6.8: How much extra fuel does it take if another passenger
(including luggage) is added to the plane for a total mass gain of 175 lbm? Assume the
flight time is 5.25 hours and the SFC is constant during level cruising at 0.587
(lbm/hr)/(lbf of thrust). The original thrust was 21,600 lbf.
Need: Extra fuel required for an extra passenger of mass 175 lbm.
Know: Original condition of plane:
Mass of plane + passengers = 324,000 lbm,
Thrust = 21,600 lbf;
Total lift to drag = 15.0
SFC was and remains at 0.587 (lbm/hr)/(lbf of thrust).
Fuel consumption = 12,680 lbm/hr.
Flight time was 5.25 hours so the total fuel consumed was 66,570 lbm.
3 Exclusive of the cost to the airline of a meal of peanuts!
15. A new development in airfoil design is the so called “scimitar winglet” as seen below
on a United Airlines airplane.
Actually this particular picture shows a double scimitar winglet, one facing upward and
one facing downward. Boeing states these winglets save several percent in drag (and thus
fuel) by reducing the size of vortices shed from the wingtip in normal flight. On a Boeing
737 these winglets stand as much as 3.5 m over the wingtip surfaces.
What’s the net improvement in lift-over-drag ratio given the unmodified plane had a lift-
to-drag ratio of 17.0 while the drag has been reduced by 4.00%?
Your solution must also compensate for the combined mass of 220.0 lbm of two winglets.
Assume the unmodified weight was 176,000 lbf.
Need: By adding scimitars to an existing aircraft, calculate the
improvement in its L/D ratio over a standard wing foil.
Know: When plane is in steady level flight L/D = 17.0 and lift = weight =
176,000 lbf. Also numerically lbm = lbf.
Kosky, Balmer, Keat and Wise: Exploring Engineering, Fourth Edition
16. Using the results of Exercise 15 what is the reduction in fuel rate of consumption in
steady level flight? Assume the fuel rate is proportional to the thrust when in steady level
flight.
Need: How much fuel is offset by scimitar winglets assuming the fuel use
is proportional to the thrust?
17.7.
17. Nevil Shute was a British/Australian author and an aeronautical engineer and highly
successful in both careers. He wrote an influential book entitled “No Highway” about a
post-World War II British airliner he called the “Reindeer”4.
The narrator is a hard driving government employed aeronautical engineer, Dr. Scott,
who takes over a laboratory staffed by over-the-hill specialists, most of whom he quickly
retires. But one metallurgist, Mr. Honey, is a quiet and unassuming scientist (but iron
willed). He is working on stress fracture5 of the tail plane of the Reindeer. Mr. Honey
says the tail unit will catastrophically fail in a certain number of flight hours and Dr.
Scott wonders if Mr. Honey is just another over-the-hill specialist. In fact a full scale
experiment of the Reindeer tail section of Mr. Honey’s is already underway to test for
stress fractures. Mr. Honey’s projection for a catastrophic failure is already overdue.
Then a Reindeer crashes killing all on board in remote Canada (while unfortunately
carrying a Soviet ambassador and causing an international incident).
Could Mr. Honey have been right all along? Should Dr. Scott have grounded the
Reindeer when he first heard of Mr. Honey’s theory? Or was his impression of Mr.
Honey as ineffectual correct?
Is there an ethical dilemma here? If Dr. Scott thinks Mr. Honey is wrong why has Mr.
Honey been allowed to waste the government’s money on a large and useless
experiment? On the other hand, if Dr. Scott believes Mr. Honey, why didn’t he ground
the Reindeer immediately?
Use an Engineering Ethics Matrix with these top line choices for Dr. Scott (you can add
more of your own).
4 The world’s first passenger jet was the Comet. (Reindeer and Comet seem to be related by a well-known
Santa poem!) The Comet did suffer from stress fatigue cracking and several aircraft went down with 100%
fatalities by the time the problem was fixed.
5 Take a piece of a tin can and bend it back and fro. It will eventually crack. This is a stress fracture.
Kosky, Balmer, Keat and Wise: Exploring Engineering, Fourth Edition
Actions for
Dr.Scott
Canons
Fire Mr.
Honey.
Encourage
Mr. Honey’s
expt.
Ground all
Reindeers
until
fracture
possibility
ruled out
Effects on
British
aeronautical
engineering
Report to a
newspaper
reporter
1) Hold
paramount the
safety, health
and welfare of
the public.
No. Mr. Honey
might be right
The only
way for this
is to keep the
expt. running
Yes.
“Paramount”
allows no
discussion.
Ground the
Reindeers.
-
2) Perform
services only in
the area of your
competence
Yes. This is
well within Dr.
Scott’s
purview
Yes, within
his
competence
Yes
Not really
qualified to
discuss
Yes, you
might for
the
government
3) Issue public
statements only
in an objective
and truthful
manner
Yes. Within
Dr. Scott’s
responsibilities
Yes, within
Dr. Scott’s
competence
Yes
Not qualified
Yes, you
might
through the
government
4) Act for each
employer or
client as
faithful agents
or trustees
Yes, but you
are acting for
the
government,
not the airline
nor the aircraft
manufacturer
Unclear if
encouraging
Mr. Honey
or stopping
him is the
correct path
Not clear
which
action to
take.
-
Yes, acting
for the
government.
5) Avoid
deceptive acts
This is only
deceptive if
you “ground”
Mr. Honey and
let the
Reindeer fly.
Not
deceptive
Not
deceptive
Not deceptive
Not
deceptive if
through the
government
6) Conduct
themselves
honorably …
Saving lives is
honorable but
what do you
owe the
airlines and
your
government?
Has the
potential for
saving lives.
If on best
available
data,
either
course is
honorable.
If and only if
Reindeer is
grounded and
Mr. Honey
continues
expt.
Yes you
might
through the
government
